reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;

theorem Th4:
  for X being BCK-algebra,x being Element of X holds (0.X,x) to_power n = 0.X
proof
  let X be BCK-algebra;
  let x be Element of X;
  defpred P[Nat] means $1<= n implies (0.X,x) to_power $1 = 0.X;
  now
    let k;
    assume
A1: k<= n implies (0.X,x) to_power k = 0.X;
    set m = k+1;
    assume m<=n;
    then (0.X,x) to_power m = x` by A1,BCIALG_2:4,NAT_1:13
      .= 0.X by BCIALG_1:def 8;
    hence (0.X,x) to_power m = 0.X;
  end;
  then
A2: for k st P[k] holds P[k+1];
A3: P[0] by BCIALG_2:1;
  for n holds P[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
